Skip to main content
SpringerLink
Log in

Heuristic Methods for Linear Multiplicative Programming

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

The linear multiplicative programming is the minimization of the product of affine functions over a polyhedral set. The problem with two affine functions reduces to a parametric linear program and can be solved efficiently. For the objective function with more than two affine functions multiplied, no efficient algorithms that solve the problem to optimality have been proposed, however Benson and Boger have proposed a heuristic algorithm that exploits links of the problem with concave minimization and multicriteria optimization. We will propose a heuristic method for the problem as well as its modification to enhance the accuracy of approximation. Computational experiments demonstrate that the method and its modification solve randomly generated problems within a few percent of relative error.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Avriel, W.E. Diewert, S. Schaible and I. Zang (1988), Generalized Convexity, New York: Plenum Press.

    Google Scholar 

  2. H.P. Benson and G.M. Boger (1997), Multiplicative programming problems: analysis and efficient point search heuristic, Journal of Optimization Theory and Applications 94: 487-510.

    Google Scholar 

  3. F. Forgo (1975), The solution of special quadratic programming problem, SZIGMA 8: 53-59 (in Hungarian).

    Google Scholar 

  4. F. Forgo (1988), Nonconvex Programming, Budapest, Hungary: Akademiai Kiado.

    Google Scholar 

  5. H. Konno, T. Kuno and Y. Yajima (1992), Parametric simplex algorithm for a class of NPcomplete problems whose average number of steps is polynomial, Computational Optimization and Applications 1: 227-239.

    Google Scholar 

  6. H. Konno and T. Kuno (1992), Linear mutliplicative programming, Mathematical Programming 56: 51-64.

    Google Scholar 

  7. H. Konno and T. Kuno (1995), Multiplicative programming, in: R. Horst and P.M. Pardalos (eds.), Handbook of Global Optimization (pp. 369-405) Dordrecht, Boston/London: Kluwer Academic Publishers.

    Google Scholar 

  8. T. Kuno (1999), A finite branch-and-bound algorithm for linear multiplicative programming, ISE-TR-99-159. Institute of Information Sciences and Electronics, University of Tsukuba.

    Google Scholar 

  9. T. Kuno, Y. Yajima and H. Konno (1993), An outer approximation method for minimizing the product of several convex functions on a convex set, Journal of Global Optimization 3: 325-335.

    Google Scholar 

  10. T. Matsui (1996), NP-hardness of linear multiplicative programming and related problems, Journal of Global Optimization 9: 113-119.

    Google Scholar 

  11. P.M. Pardalos and S.A. Vavasis (1991), Quadratic progamming with one negative eigenvalue is NP-hard, Journal of Global Optimization 1: 15-22.

    Google Scholar 

  12. S. Park (1997) LPAKO ver4.0f code for Linear Programming Program, Seoul National University.

  13. H.S. Ryoo and N.V. Sahinidis (1996), A branch-and-reduce approach to global optimization, Journal of Global Optimization 8: 107-138.

    Google Scholar 

  14. R.E. Steuer (1986), Multiple Criteria Optimization: Theory, Computation, and Application. New York: John-Wiley and Sons.

    Google Scholar 

  15. R.E. Steuer (1994), Random problem generation and the computation of efficient extreme points in multiple objective linear programming, Computational Optimization and Applications 3: 333-347.

    Google Scholar 

  16. R.E. Steuer (1995),Manual for the ADBASE: Multiple Objective Linear Programming, mimeo.

  17. K. Swarup (1966a), Programming with indefinite quadratic function with linear constraints, Cahiers du Centre d'Études de Recherche Opérationnelle 8: 132-136.

    Google Scholar 

  18. K. Swarup (1966b), Indefinite quadratic programming, Cahiers du Centre d'Études de Recherche Opérationnelle 8: 217-222.

    Google Scholar 

  19. K. Swarup (1966c), Quadratic programming, Cahiers du Centre d'Études de Recherche Opérationnelle 8: 223-234.

    Google Scholar 

  20. N.V. Thoai (1991), A global optimization approach for solving the convex multiplicative programming problem, Journal of Global Optimization 1: 341-357.

    Google Scholar 

  21. H. Tuy (1991), Polyhedral annexation, dualization and dimension reduction technique in global optimization, Journal of Global Optimization 1: 229-244.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Policy and Planning Sciences, University of Tsukuba, Tsukuba, Ibaraki, 305-8573, Japan

    X.J. Liu, T. Umegaki & Y. Yamamoto

Authors
  1. X.J. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Umegaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Y. Yamamoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, X., Umegaki, T. & Yamamoto, Y. Heuristic Methods for Linear Multiplicative Programming. Journal of Global Optimization 15, 433–447 (1999). https://doi.org/10.1023/A:1008308913266

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008308913266

Keywords

Search

Navigation